IB Physics Sub-topic D1 Notes

Gravitational potential

In Topic B.5, you learned that electric fields have a potential difference. The same applies to any other field, including gravitational fields. Thus, gravitational potential (V_g) is the energy per unit test point mass, measured in J kg^-1. Newton’s law of universal gravitation provides a formula for the gravitational potential at a distance (r) from any point mass (M):

$V_{g} = \frac{-GM}{r}$

Remember that when a test point is moved:

1. Parallel to field lines - the test point mass is moving in the direction of the force. Therefore, work is done on the object.
2. Perpendicular to field lines - the test point mass is not moving in the direction of the force. Therefore, no work is done on the object.

When work is done moving a test point mass between two locations, the energy is transferred into changing its potential. Therefore, there is a gravitational potential difference (ΔV) between the two locations. The gravitational potential difference (ΔV_g) is defined as the work done moving a test point mass per unit test point mass, measured in J kg^-1.

If work is done on the test point mass to change its potential, the potential difference will be positive. If the test point mass does work itself to change its potential, the potential difference will be negative. The formula for this is:

$W = m\Delta V_{g}$

Whenever the potential difference is measured, it assumed that the gravitational field strength is constant and that the gravitational potential energy at the surface is zero.

Since this is not the case, infinity is the point of reference considered the zero of gravitational potential energy. As such, gravitational potential energy (E_p) is defined as the work done in moving a mass from infinity to any point in space.

The formula for this is:

$E_{p} = - \frac{GMm}{r}$

Equipotential

However, when no work is done in moving an object in a field, the locations must have no potential difference. These locations are said to be equipotential. Orbits are an example of an equipotential location.

Equipotential lines are represented as lines perpendicular to the field lines, since that is the plane of motion wherein no work is done moving a test point.

For example, if the field diagram is:

The equipotential diagram is: